Monodromy of four‐dimensional irreducible compatible systems of Q$\mathbb {Q}$

Abstract

Let 𝐹 be a totally real field and 𝑛 ⩽ 4 a natural number. We study the monodromy groups of any 𝑛-dimensional strictly compatible system {𝜌 𝜆 } 𝜆 of 𝜆-adic representations of 𝐹 with distinct Hodge-Tate numbers such that 𝜌 𝜆 0 is irreducible for some 𝜆 0 . When 𝐹 = ℚ, 𝑛 = 4, and 𝜌 𝜆 0 is fully symplectic, the following assertions are obtained.(i) The representation 𝜌 𝜆 is fully symplectic for almost all 𝜆. (ii) If in addition the similitude character 𝜇 𝜆 0 of 𝜌 𝜆 0 is odd, then the system {𝜌 𝜆 } 𝜆 is potentially automorphic and the residual image ρ𝜆 (Gal ℚ ) has a subgroup conjugate to Sp 4 (𝔽 𝓁 ) for almost all 𝜆.M S C 2 0 2 0 11F80, 11F70, 11F22, 20G05 INTRODUCTION Main resultsLet 𝐾, 𝐸 be number fields, and {𝜌 𝜆 } 𝜆 an 𝐸-rational compatible system of semisimple 𝑛dimensional 𝜆-adic representations of 𝐾 in the sense of Serre (Definition 2.2). Denote by Γ 𝜆 the monodromy (i.e., image) of 𝜌 𝜆 and by 𝐆 𝜆 the algebraic monodromy group of 𝜌 𝜆 † . A fundamental † The Zariski closure of the monodromy Γ 𝜆 in GL 𝑛,𝐸 𝜆 .In honor of Professor Michael Larsen's 60th birthday.